Optimal. Leaf size=442 \[ -\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {32 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d}+\frac {32 e^3 \sqrt {2 \pi } S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{15 b^{7/2} d} \]
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Rubi [A]
time = 0.72, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4889, 12,
4729, 4807, 4727, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {32 \sqrt {2 \pi } e^3 \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 \sqrt {\pi } e^3 \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 \sqrt {\pi } e^3 \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}+\frac {32 \sqrt {2 \pi } e^3 \sin \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {128 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{15 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}-\frac {16 e^3 \sqrt {1-(c+d x)^2} (c+d x)}{5 b^3 d \sqrt {a+b \text {ArcSin}(c+d x)}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d (a+b \text {ArcSin}(c+d x))^{3/2}}-\frac {2 e^3 \sqrt {1-(c+d x)^2} (c+d x)^3}{5 b d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4727
Rule 4729
Rule 4807
Rule 4889
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \sin ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \sin ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}+\frac {\left (6 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}-\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {\left (16 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (128 e^3\right ) \text {Subst}\left (\int \left (\frac {\cos (2 x)}{2 \sqrt {a+b x}}-\frac {\cos (4 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}-\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (16 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (16 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {\left (32 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac {\left (64 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac {\left (64 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{5 b^{7/2} d}+\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{5 b^{7/2} d}-\frac {\left (128 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (128 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}-\frac {\left (128 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (128 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{15 b^4 d}\\ &=-\frac {2 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{5 b d \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}-\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x) \sqrt {1-(c+d x)^2}}{5 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {128 e^3 (c+d x)^3 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \sin ^{-1}(c+d x)}}+\frac {32 e^3 \sqrt {2 \pi } \cos \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {16 e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d}+\frac {32 e^3 \sqrt {2 \pi } S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{15 b^{7/2} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.57, size = 445, normalized size = 1.01 \begin {gather*} \frac {e^3 \left (-4 (a+b \text {ArcSin}(c+d x)) \left (e^{2 i \text {ArcSin}(c+d x)} (4 i a+b+4 i b \text {ArcSin}(c+d x))+4 \sqrt {2} b e^{-\frac {2 i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{-2 i \text {ArcSin}(c+d x)} \left (-4 i a+b-4 i b \text {ArcSin}(c+d x)+4 \sqrt {2} b e^{\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )+4 (a+b \text {ArcSin}(c+d x)) \left (e^{-4 i \text {ArcSin}(c+d x)} (-8 i a+b-8 i b \text {ArcSin}(c+d x))+e^{4 i \text {ArcSin}(c+d x)} (8 i a+b+8 i b \text {ArcSin}(c+d x))+16 b e^{-\frac {4 i a}{b}} \left (-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},-\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )+16 b e^{\frac {4 i a}{b}} \left (\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )^{3/2} \text {Gamma}\left (\frac {1}{2},\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )-6 b^2 \sin (2 \text {ArcSin}(c+d x))+3 b^2 \sin (4 \text {ArcSin}(c+d x))\right )}{60 b^3 d (a+b \text {ArcSin}(c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1264\) vs.
\(2(368)=736\).
time = 0.53, size = 1265, normalized size = 2.86
method | result | size |
default | \(\text {Expression too large to display}\) | \(1265\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{3} \left (\int \frac {c^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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